Intuitive Set Theory
K. K. Nambiar
6664 Autumn Glen Drive, West Chester, OH 45069
(Received December 1998)
Abstract—A set theory called Intuitive Set Theory is introduced in which Skolem Paradox
does not appear. A measure function called real measure is defined in which Axiom of Choice
cannot produce a nonmeasurable set.
Keywords—Skolem Paradox; Bonded set; Real measure.
1. INTRODUCTION
Intuitive Set Theory (IST) defined here is an extension of the Real Set Theory (RST) given in
an earlier paper [1], in the sense that the Axiom of Fusion defined below holds good in IST, in
addition to all the axioms of RST. The concept of a class of bonded sets introduced here allows
us to construct a model [2] of IST with real cardinality ℵ0 , thereby avoiding the Skolem Paradox.
The notion of a bonded set is useful also for defining a real measure in which the Axiom of Choice
(AC) cannot be used to produce a nonmeasurable set.
2. INTUITIVE SET THEORY
Some definitions are given below to facilitate our discussion. The crucial concept here is a
bonded set, a set which even the axiom of choice cannot penetrate to choose an element.
Bonded Set: A set is a bonded set, if the AC can choose only the set itself as a whole and
not its constituent elements.
Class: A set which has sets as its elements.
Bonded Class: A class which has only bonded sets as its elements.
2ℵα: Power set of ℵα .
ℵα
ℵα : Real power set of ℵα , defined as the class of all the subsets of ℵα of cardinality ℵα .
Real Cardinality: Cardinality of a bonded class, taking the bonded sets in it as elements.
R: The class of infinite recursive subsets of positive integers, a class of cardinality ℵ0 .
x: An element of R, which defines an infinite binary sequence and hence equivalent to a real
point in the interval (0, 1].
(x]: The same as the cartesian product x × 2ℵα , which we will call the infinitesimal x.
Microcosm: The same as the cartesian product R × 2ℵα , considered an adequate representation of all the points of (0, 1].
N : An element of R, which defines an infinite binary sequence written leftwards and hence
called a supernatural number.
[N ): The same as the cartesian product 2ℵα × N , and hence called a cosmic stretch.
Macrocosm: The same as the cartesian product 2ℵα × R, considered an adequate representation of all counting numbers, even those above supernatural numbers.
Using these definitions, we can now state the axiom that is central to IST.
Axiom of Fusion. (0, 1] = ℵℵαα = R × 2ℵα , where x × 2ℵα is a bonded set.
Thus the axiom of fusion says that the significant part of the power set of ℵα consists of ℵ0
bonded sets with cardinality 2ℵα . We accept the axiom of fusion as part of IST.
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3. INFINITESIMAL GRAPH
Note, as an example, that the infinite sequence .010 ∗ ∗ ∗ ∗ · · · can be used to represent the
interval (.25, .375], if we accept certain assumptions about the representation:
The initial binary string, .010 = .25, represents the initial point of the interval.
The length of the binary string, 3 in our case, decides the length of the interval as 2−3 = .125.
Every ∗ in the infinite ∗-string can be substituted by a 0 or 1, to create 2ℵα points in the
interval.
We will accept the fact that a nonterminating binary sequence, .bbbb · · ·, or equivalently, an
infinite recursive subset of positive integers, can be used uniquely to represent a real number in
the interval (0, 1]. A slight generalization of these ideas allows us to draw a graph to represent
[0, 1].
To visualize the definitions given above and also to aid our intuition, we define a graph
we call Gℵ0 . In the Cantorian tradition, we define the infinitesimal graph Gℵ0 as the infinite
sequence of graphs shown as Figures {G1 , G3 , G7 , · · ·}. Note that the graph Gk has k nodes
between the nodes 0 and ℵ0 , labelled 1 to k. We will take it as axiomatic that the graph Gℵ0
shown in Figure Gℵ0 has ℵ0 nodes and ℵ0 edges, and also that the edges and nodes of the graph
represent respectively the infinitesimals (x] and real points x defined earlier. In the graphs, nodes
are unconventionally drawn as vertical lines for clarity, and also in deference to Dedekind whose
cut it represents.
ℵ
0............................................................................................................................1
...............................................................................................................................0
..
..
..
Fig. G1
ℵ
0..............................................................2..............................................................1
3
...............................................................................................................................0
..
..
..
..
..
Fig. G3
ℵ
0...............................4...............................2...............................6
1
5
3
7
...............................................................................................................................................................0
..
..
..
..
..
..
..
..
..
Fig. G7
..
.
0......................................................................................................................................................................................................................................................ℵ
.....0
Fig. Gℵ0
Even though Gℵ0 has no role to play with the formal part of our arguments here, it can aid
considerably in the visualization of the concepts introduced.
4. SKOLEM PARADOX
Cantor’s theorem asserts that every model of Zermelo-Fraenkel set theory (ZF) has to have
cardinality greater than ℵ0 . On the other hand, Löwenheim-Skolem theorem (LS) says that there
is a model of ZF theory, whose cardinality is ℵ0 . These two statements together is called Skolem
Paradox.
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Intuitive set theory provides a reasonable way to resolve the Skolem Paradox. We merely
take the LS theorem as stating that the real cardinality of a model of IST need not be greater
than ℵ0 . Clearly, the Upward Löwenheim-Skolem theorem also cannot raise any paradox in IST.
5. REAL MEASURE
In measure theory, it is known that there are sets which are not Lebesgue measurable, but it
has not been possible to date to construct such a set, without invoking the axiom of choice. The
usual method is to choose exactly one element from each of the set x × 2ℵα we defined earlier,
and show that the set thus created is not Lebesgue measurable. This method is obviously not
possible in IST, since x × 2ℵα is a bonded set. Noting this fact, we define real measure in IST
to be the same as Lebesgue measure, except that the universal set we consider is restricted to
the interval [0, 1]. If we accept the axiom of fusion, it is easy to see that the axiom of choice
cannot be used to produce a nonmeasurable subset of [0, 1]. Thus, it would not be unreasonable
to assert that there are no sets in IST which are not real measurable.
6. CONCLUSION
From the definitions given in the beginning, it should be obvious that there is a one-to-one
correspondence between the points in (0, 1] and the counting numbers. Hence, our statements
about microcosm are equally applicable to macrocosm also. Further, it should be clear that intuitive set theory will suffice for scientists to investigate the phenomenal world, and real set theory
will be needed only by mathematicians who want to probe the complexities of the noumenal
universe.
REFERENCES
1. K. K. Nambiar, Real Set Theory, Computers and Mathematics, Vol. ??, No. ?, pp. ?-?, (199?).
2. P. Benacerraf, and H. Putnam, eds., Philosophy of Mathematics: Selected Readings, Prentice-Hall, Englewood
Cliffs, NJ, (1964).
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